The set of rational numbers, \(\mathbb{Q}\) is defined as \[\left\{ \frac{p}{q} \mid p \in \mathbb{Z} \text{ and } q \in \mathbb{Z} \text{ and } q \neq 0 \right\} \text{~~~~or, equivalently,~~~~} \{ x \in \mathbb{R} \mid \exists p \in \mathbb{Z} \exists q \in \mathbb{Z}^+ ( p = x \cdot q) \}\] Extra practice: Use the definition of set equality to prove that the definitions above give the same set.